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This paper was written while the third auther was a guest of the research project SFB 237 of the DFG at the Ruhr University Bochum. The first auther was also supported by the DFG.

1. The Seiberg-Witten equation In order to be able to write down the equation of Seiberg-Witten, we need to recall the definition of a Spinc-structure. (For details see [LM]). Let (X,g) be an oriented, compact Riemannian 4-manifold and PSO(4) → X c c its oriented orthonormal frame bundle. Spin (4) = Spin(4) ×Z2 U(1). A Spin - structure is a lift of the structure group SO(4) to Spinc(4), i.e. there exists a c principal Spin -bundle PSpinc(4) → X such that there is a bundle map

PSpinc(4) −→ PSO(4)

↓ ↓

X −→ X

Let Q = PSpinc(4)/Spin(4) be a principal U(1)-bundle. W = PSpinc(4) ×Spinc(4) 4 C and L = Q ×U(1) C resp. is the associated spinor bundle and the line bundle resp.. W can be decomposed globally as W + and W −. Locally,

W ± = S± ⊗ L1/2.

Here S± is a spinor bundle w. r. t. a local Spin-struture on X. Both S± and L1/1 are locally defined. There exists a Clifford multiplication

TX × W + → W −

denoted by e · φ ∈ W − for e ∈ TX and φ ∈ W +. Here TX is the tangent bundle of X. A connection on the bundle W + can be defined by the Levi-Civita connection + − and a connection on L. The “twisted” Dirac operator DA : Γ(W ) → Γ(W ) is defined by 4 DA = ei ·∇A. i=1 X ± ± Here, Γ(W ) is the space of sections of W , {ei} is an orthonormal basis of TX and + ∇A is a connection on W induced by the Levi-Civita connection and a connection A on the line bundle L. Definition 1.1. The Seiberg-Witten equations are

DAφ =0, (1.1) i F + = he e φ, φiei ∧ ej , A 4 i j

+ for A a connection on L and φ ∈ Γ(W ), where F (A) = −iFA is the curvature of + i A, FA is the self dual part of FA and {e } is the dual basis of {ei}. VARIATIONAL ASPECTS OF THE SEIBERG-WITTEN FUNCTIONAL 3

Definition 1.2. The Seiberg-Witten functional is

i (1.2) SW (A, φ) = (|D |2 + |F + − he e φ, φiei ∧ ej|2)dvol A A 4 i j ZX

The Euler-Lagrange equations of the Seiberg-Witten functional are

∗ i + 1 DADAφ − FA · φ − heiej φ, φieiejφ =0, (1.3) 2 8 i 1 d∗(F + − he e φ, φiei ∧ ej) + ImhD φ, e φiei =0. A 4 i j 2 A i

∗ ∗ Here, DA is the formal adjoint operator of DA, d = −∗ d∗ and ∗ is the Hodge star operator. It is easy to see that a solution of (1.1) is a solution of (1.3). In fact, it is clear from the definition of the functional that a solution of (1.1) is a minimizer of the Seiberg-Witten equation. The following Weitzenb¨ock formula plays an important role in the Seiberg-Witten theory,

s i (1.4) D∗ D φ = −∆ φ + φ + F · φ, A A A 4 2 A where s is the scalar curvature of (X,g). By this formula, the Seiberg-Witten functional can be rewritten as s 1 (1.5) SW (A, φ) = (|∇ φ|2 + |F +|2 + |φ|2 + |φ|4)dvol. A A 4 8 ZX And the Euler-Lagrange equation (1.3) can be rewritten as

s 1 2 −∆Aφ + φ + |φ| φ =0, (1.6) 4 4 1 d∗F + + Imh∇ φ, φiei =0. A 2 i

Here ∆A is the analyst’s Laplacian, the negative Laplacian, and ∇i = ∇ei . Lemma 1.3. For a smooth solution (A, φ) of equation (1.3) (or (1.6)), |φ|(x) ≤ max{−s0, 0}, where s0 = min{s(x)|x ∈ X}. Proof. From the maximum principle.  Corollary 1.4. If the scalar curvature of X is nonnegative, then for any smooth solution (A, φ) of (1.2), φ ≡ 0. Hence, a solution of (1.2) shares many properties with solutions of the Seiberg- Witten equation (1.1). In section 3 below, we shall prove Lemma 1.3 for weak solutions of (1.2). In this paper we shall consider the Seiberg-Witten functional and prove a com- pactness theorem. The precise set-up of the problem will be given in section 2. Before ending the present section, we want to give the definition of the Palais- Smale condition. 4 JURGEN¨ JOST, XIAOWEI PENG AND GUOFANG WANG

Let M be a Banach manifold and f : M → R a smooth functional. Let G be a Lie group acting on M and suppose f is invariant under G, i.e. for any g ∈ G and x ∈ X, f(gx) = f(x). f is said to satisfy the Palais-Smale condition: if for any sequence {xi}i∈N with (i) f(xi) is bounded, (ii) df(xi) → 0, as i →∞, there exists a subsequence (also denoted by {xi}) and a sequence gi ∈ G such that gixi converges in X to a critical point x of f, i.e. df(x) = 0, with f(x) = limi→∞ f(xi). 2. The set-up We need to choose a suitable working space to discuss the Seiberg-Witten func- tional. For a vector bundle with a metric over X, we define Lk,p(E), the Sobolev space of sections of E by completing the space Γ(E) of smooth sections of E by k α p 1/p kskLk,p =( |∇ s| ) , |αX|=0 Z where ∇ is a fixed metric connection on E. (For details see [P]). For such Sobolev spaces, the Sobolev embedding theorem and the H¨older inequality are valid, i.e.

4 1/4 1,2 (2.1) ( |s| ) ≤ ckskL1.2 , for s ∈ L , ZX and 2 2 1/2 4 1/4 4 1/4 (2.2) ( |s1| |s2| ) ≤ ( |s1| ) ( |s2| ) . ZX ZX ZX Let L1,2(W +) be the space of sections of the bundle W + of class L1,2 defined above and A1,2 = L1,2(A) the space of connections of class L1,2 defined by a fixed 1,2 + 1,2 connection A0. L (W ) and A are Hilbert manifolds. Lemma 2.1. The Seiberg-Witten functional SW is well defined on A1,2×L1,2(W +) and SW is smooth. Proof. From (2.1) and (2.2),

2 |Aφ| ≤ ckAkL1,2 kφkL1,2 . ZX This implies that SW is well-defined. It is easy to check that SW is smooth.

Now let us to choose a suitable Lie group as a gauge group. First let G0 = 2,2 exp(iL (X, R)). We claim that G0 is a Lie group. Actually, G0 can be seen as 2,2 a quotient of L (X, R) under an equivalence relation ∼. φ1 ∼ φ2 if and only if φ1(x) − φ2(x)=2πn, for almost all x ∈ X, for some integer n. It is clear that 2,2 Y = L / ∼ is a Lie group with the usual addition of functions. G0 can be identified with Y by the exponential map. Hence G0 is a Lie group with the multiplication of functions, as the identity component in the continuous case. Then by a well- known result about harmonic maps from X into S1 ([EL]), in any component of C∞(X, S1) there exists a unique map g ∈ C∞(X, S1) such that (2.3) d∗(g−1dg)=0,

(2.4) g(x0)=1, for a fixed point x0 ∈ X. VARIATIONAL ASPECTS OF THE SEIBERG-WITTEN FUNCTIONAL 5

Lemma 2.2. Let G = ∪g · G0. G is a Lie group, where the union is over all components of C∞(X, S1).

Proof. Since G0 is a Lie group, it suffices to check the following two points. ∞ 1 (i) For two different components of C (X, S ) with as g1 and g2 obtained above, g1G0 ∩ g2G0 = ∅. Assuming that g1G0 ∩ g2G0 6= ∅, there exist ϕ1 and ϕ2 ∈ G0 such that g1ϕ1 = g2ϕ2, for almost all x ∈ X. From the definition of G0, there exist ζ1 and ζ2 ∈ L2,2(X, R) such that −1 i(ζ2−ζ1) g1g2 = e , for almost all x ∈ X. From (2.3) above, we have ∗ d d(ζ2 − ζ1)=0, 2,2 i.e. ζ2 −ζ1 ∈ L (X, R) is a harmonic function. Hence tegether with (2.4), ζ2 = ζ1. Therefore g1 = g2, a contradiction. (ii) The operation of the group is closed. Let g1ϕ1 ∈ g1G0, g2ϕ2 ∈ g2G0, then

g1ϕ1 · g2ϕ2 = g1g2ϕ1ϕ2 ∈ g1g2G0 and g1g2 satisfies (2.3) and (2.4) and is the corresponding g of some component of C∞(X,R) as above.  Lemma 2.3. G acts smoothly on A1,2 × L1,2(W +). Proof. G acts on A1,2 × L1,2(W +) as follows g(A, φ)=(g(A),g−1φ), for (A, φ) ∈ A1,2 × L1,2(W +) and g ∈ G, where g(A) = A + g−1dg. It is easy to check that the action is well-defined and smooth.  Remark 2.4. In the non-Abelian case, an element g of the gauge group of class L2,2 need not act smoothly on A1,2. So on A1,2 ×L1,2(W +), we can consider the Seiberg-Witten functional. It is easy to check that (1.2) is equivalent to (1.5) by an approximation argument. Here, we prefer to use the form (1.5). Lemma 2.5. The Seiberg-Witten functional SW is coercive, i.e., there exists a constant c > 0 such that for each (A, φ) ∈A1,2 × L1,2(W +)

−1 −1 SW (A, φ) ≥ c (kg φkL1,2 + kg(A)kL1,2 ) − c, for some g ∈ G. Proof. First, we have s 1 SW (A, φ) = (|∇ φ|2 + |F +|2 + |φ|2 + |φ|4) A A 4 8 Z 1 = {(|∇ φ|2) + |φ|4 A 16 Z 1 1 + (4s2 +4s|φ|2 + |φ|4) − s2} 16 4 1 ≥ (|∇ φ|2 + |F +|2 + |φ|4) − c. A A 16 Z 6 JURGEN¨ JOST, XIAOWEI PENG AND GUOFANG WANG

Using the H¨older inequality, we have

(2.5) ( |φ|2)1/2 ≤ c( |φ|4)1/4 ≤ c |φ|4 + c. Z Z Z In this paper, c is a constant that may change from term to term. Also, we have

2 2 |∇φ| ≤ |∇Aφ − (A − A0)φ| ZX Z 2 2 2 ≤ 2 (|∇Aφ| + |A − A0| |φ| ) Z 2 4 1/2 4 1/2 ≤ 2 |∇Aφ| + 2( |A − A0| ) ( |φ| ) . Z Z Z Together with estimate (2.10) below, it follows that

2 2 2 (2.6) |∇φ| ≤ c |∇Aφ| + c |FA| + c. Z Z Z Now we esitmate the term containing g(A). There exists a standard method for the Abelian case. For convenience, we give a complete proof. Step 1 (gauge fixing). There exists g0 ∈ G0 such that

∗ (2.7) d (g0(A) − A0)=0.

iζ 2,2 Since g0 = e for some ζ ∈ L (X, R), (2.7) is equivalent to

∗ ∗ (2.8) d dζ = id (A − A0).

∗ 1,2 (2.8) is solvable, for X d (A − A0) = 0 and A ∈ L . By the elliptic estimate, we have R −1 −1 c (kAkL1,2 + kφkL1,2 ) ≤kg(A)kL1,2 + kg φkL1,2 ≤ c(kAkL1,2 + kφkL1,2 ),

∗ for some constant c. Hence for simplicity, we denote g(A) by A. So d (A−A0)=0. Step 2 (component fixing). The component group of C∞(X, S1) is isomorphic to H1(X, Z). For any component of C∞(X, S1), there exists a map g satisfying (2.3) and (2.4). We know −1 g(A) − A0 = g dg + A − A0.

The harmonic part of g(A) − A0 is the harmonic part of A − A0 plus the harmonic part of g−1dg. Since the harmonic part of g−1dg belongs to H1(X, Z) and the Jacobi torus H1(X, R)/H1(X, Z) is compact, we can choose a component such that ∗ the harmonic part of g(A)−A0 is bounded. Since g is harmonic, d (g(A)−A0)=0. Step 3. Using the Hodge decomposition, we have

∗ kg(A) − A0kL1,2 ≤ c(kd (g(A) − A0)kL2 + kd(g(A) − A0)kL2 + kH(g(A) − A0)kL2 ), VARIATIONAL ASPECTS OF THE SEIBERG-WITTEN FUNCTIONAL 7 where H(g(A) − A0) is the harmonic part of g(A) − A0. From the preceding dis- cussion, we obtain

kg(A) − A0kL1,2 ≤ ckd(g(A) − A0)kL2 + c

(2.9) ≤ ckd(g(A))kL2 + c

≤ ckFAkL2 + c.

Therefore,

1 2 2 + 2 (2.10) kg(A) − A0kL , ≤ ckFAkL + c ≤ ckFA kL + c,

+ 2 − 2 the last inequality is from that kFA kL2 −kFA kL2 is independent of A. Now (2.6) and (2.10) imply

−1 −1 1 2 1 2  SW (A, φ) ≥ c (kg φkL , + kg(A)kL , ) − c.

3. Regularity of weak solutions As in the Yang-Mills case, for the Seiberg-Witten equations there is some kind of removing singularity theorem (see [Z] and for Yang-Mills see [FU] and [U]). Actually, we shall prove in this section that all weak solutions of the second order equations (1.3) are smooth. This result will be used in the proof of the Main Theorem. Theorem 3.1. Let (A, φ) ∈A1,2 ×L1,2(W +) be a weak solutin of (1.3), i.e. (A, φ) is a critical point of the Seiberg-Witten functional. Then there exists a gauge trans- fomation g ∈ G such that g(A, φ)=(g(A),g−1φ) is smooth.

First, we show the boundedness of kφkL∞ for a weak solution. Lemma 3.2. Let (A, φ) ∈A1,2 × L1,2(W +) be a weak solution of (1.3). Then

kφkL∞ ≤ max{− min s(x), 0}. x∈X

Proof. We use the method of Taubes [T3] to prove the lemma. Let s0 = minx∈X s(x). If s0 ≥ 0, then from the first equation of (1.6), (recall that (1.3) and (1.6) are equivalent) we have

s 1 |∇ φ|2 + |φ|2 + |φ|4 =0, A 4 4 Z it follows that φ = 0. So we may assume s0 = −1. Define a test section η ∈ L1,2(W +) by

(|φ| − 1) φ , for |φ| > 1, (3.1) η = |φ| ( 0, for |φ| ≤ 1.

Let ν = φ/|φ| for |φ| ≥ 1. It is clear that |ν| = 1 and

∇η =(d|φ|)ν +(|φ| − 1)∇ν. 8 JURGEN¨ JOST, XIAOWEI PENG AND GUOFANG WANG

Since (A, φ) is a weak solution of (1.6), we have

s 1 0 = h∇ φ, ∇ ηi + hφ, ηi + |φ|2hφ, ηi A A 4 4 ZΩ 1 = h∇ φ, ∇ ηi + (|φ|2 + s)(|φ| − 1)|φ| A A 4 ZΩ 1 ≥ h∇ , ∇ ηi + (|φ|2 − 1)(|φ| − 1)|φ|, A A 4 ZΩ for s ≥ −1. Here, Ω = {x ∈ X||φ|(x) > 1}. The second term of the last integration is nonnegative. And the first term is also nonnegative. In fact, we have, for x ∈ Ω,

h∇Aφ, ∇Aηi = h∇Aφ, d|φ|νi + h∇Aφ, (|φ| − 1)∇Aνi 2 2 2 2 =(|φ| − 1)|∇Aν| + h∇Aφ, νi +(|φ| − 1) |∇Aν| (3.2) +(|φ| − 1)h∇Aφ, νihν, ∇Aνi 1 ≥ h∇ φ, νi2 +(|φ| − 1)|∇ ν|2 2 A A ≥ 0, it follows that the set Ω has measure zero, in other words,

kφkL∞ ≤ 1.

In the general case, i.e. without the normaliztion s0 = −1, the prceding arguments imply kφkL∞ ≤ max{−s0, 0}. This completes the proof of the lemma.  The proof of the Theorem 3.1 is now easy: We are assuming that (A, φ) is a critical point of SW , and thus that SW (A, φ) −1 is bounded. Lemma 2.5 then implies bounds for kg φkL1,2 and kg(A)kL1,2 . Here we also denote (g(A),g−1φ) by (A, φ). Next, we have

∗ ∗ + A i (3.3) d FA =2d FA = −Imh∇i φ, φie (from (1.6))

Since kφkL∞ is bounded by Lemma 3.1, this implies

∗ (3.4) kd FAkL2 ≤ c(k∇φkL2 + kAkL2 ).

∗ The second Bianchi identity dFA = 0 and the ellipticity of d + d imply

∗ (3.5) kFAkL1,2 ≤ c(kd FAkL2 + kFAkL2 )

(3.4), (3.5) and the L1,2 estimate for A yield

(3.6) kAkL2.2 ≤ c, VARIATIONAL ASPECTS OF THE SEIBERG-WITTEN FUNCTIONAL 9 and by the Sobolev embedding theorem then also

(3.7) kAkLr ≤ c, for any r < ∞.

From (1.6), we get , for 1 < p < 2

2 k∆φkLp ≤ c(k∆AφkLp + k∇AkLp + k|A||∇φ|kLp + k|A| kLp ),

and from the H¨older inequality

k|a||∇φ|kLp ≤kAk 2r k∇φkL2 . L 2−p

4 1, p Thus, by Sobolev’s embedding theorem again, φ ∈ L 4−p and we may then apply the same kind of argument also for p = 2 and get

kφkL2,2 ≤ c.

The standard bootstrap argument then gives Lk,2 bounds for (A, φ) for any k ≥ 2, hence smoothness. 

4.The compactness theorem Main Theorem. The Seiberg-Witten functional SW satisfies the following Palais- Smale condition: 1,2 1.2 + For any sequence (An, φn) ∈A × L (W ) satisfying −1,2 −1,2 + (i) dSW (An, φn) → 0 strongly in A × L (W ); (ii) SW (An, φn) ≤ c, for n =1, 2,...,

there exists a subsequence (denoted by (An, φn)) and gn ∈ G such that gn(An, φn) converges in A1,2 × L1,2(W +) to a critical point (A, φ) of SW with SW (A, φ) = limn→∞ SW (gn(A, φ)). As we know, the crucial point in the Seiberg-Witten theory and in the preced- ing arguments is the boundedness of kφkL∞ . But for a Palais-Smale sequence (a sequence satisfying (i) and (ii)), there exists no uniform bound for kφnkL∞ . This is the main difficulty we encounter here. Fortunately, we can obtain a weaker bound from the proof of (3.2) that suffices for showing the Palais-Smale condition. Proof of Main Theorem. Step 1. By Lemma 2.5, there exist g ∈ G with

−1 kgn(An)kL1,2 + kg φkL1,2 ≤ c

−1 (independent of n). For simplicity, we denote gn(A) by An, and g φn by φn. From Rellich’s Theorem and Sobolev’s embedding theorem, there exists a sub- sequence (also denoted by (An, φn)) such that 1,2 1,2 + (i) An ⇀A weakly in A , and φn ⇀ φ weakly in L (W ). 0,4 0,4 + (ii) An ⇀A weakly in A , and φn ⇀ φ weakly in L (W ). 0,p 0,p + (iii) An → A strongly in A , p < 4, and φn → φ strongly in L (W )(p < 4). 10 JURGEN¨ JOST, XIAOWEI PENG AND GUOFANG WANG

Step 2. (A, φ) is a weak solution of (1.6). For any 1-form θ ∈A1,2 (here we abuse the notation a bit),

An i (4.1) hFAn ,dθi + Imh∇i φn, φnihe ,θi = dSW (An, φn)(θ) = o(1). Z From step 1 (i), we know

(4.2) hFAn ,dθi = hFA,dθi + o(1). Z Z Now we show that

An i A i Imh∇i φn, φnihe ,θi = Imh∇i φ, φihe ,θi + o(1) Z Z This follows from

An i A i Imh∇i φn, φnihe ,θi − Imh∇i φ, φihe ,θi Z Z i i i = he ,θi(Imh∇iφn + Anφn), φni − Imh∇iφ + A φ, φi) Z i i i i = he ,θi(Imh∇i(φn − φ) + Anφn − A φ, φni − Imh∇iφ + A φ, φ − φni) (4.3) Z i i i i = he ,θi(Imh∇i(φn − φ) + Anφ − A φ, φi − Imh∇iφn + A φ, φi) + o(1) Z (from step 1 (i) and (iii))

i i i = {ImhAn(φn − φ), φi + Imh(An − A )φ, φi}hei,θi + o(1) Z = o(1). (from step 1 (iii))

i i i i Here A − A0 = A e and An − A0 = Ane . From (4.1)–(4.3), we have for any θ,

A i hFA,dθi + Imh∇i φ, φihe ,θi =0, Z i.e. (A, φ) satisfies weakly the second equation of (1.6). Using the same argument, we can show that (A, φ) satisfies weakly the first equation of (1.6). That is, (A, φ) is a weak solution of (1.6). Hence from Theorem 3.1, there exists g ∈ G such that g(A, φ) is a smooth solution of (1.6) and |φ(x)| ≤ s0 (recall that s0 = max{− minx∈X s(x), 0}). So we may assume that (A, φ) is a smooth solution. Step 3. As in the proof of Lemma 3.2, we assume s0 := minx∈X s(x) = −1. Set

Ωn := {x ∈ X||φn| > 1},

and νn = φn/|φn| for x ∈ Ωn. VARIATIONAL ASPECTS OF THE SEIBERG-WITTEN FUNCTIONAL 11

Claim.

2 (4.4) |h∇An φn, νni| → 0, as n →∞. ZΩn

As in the proof of lemma 3.2, we have (see (3.2))

2 −1,2 1,2 |h∇An φn, νni| ≤ 2dSW (An, φn)(ηn) ≤ 2kdSW (An, φn)kL kηnkL . ZΩn

Here ηn is defined as η in the proof of Lemma 3.2, namely,

(|φ | − 1) φn , for |φ | > 1, n |φn| n ηn = ( 0, for |φn| ≤ 1.

It is sufficient to show that kηnkL1,2 ≤ c for a constant c. From the definition of ηn, we have

2 2 2 2 kηnkL2 = (|φn| − 1) ≤ |φn| + c ≤kφnkL1,2 + c ≤ c, ZΩn ZX and 2 2 2 2 k∇ηnkL = kd|φn|νn +(|φn| − 1)∇νnkL (Ωn)

2 |φ| − 1 2 ≤ 2k∇φ k 2 +8k k ∞ k∇φ k 2 n L |φ| L (Ωn) n L 2 ≤ ck∇φnkL2 ≤ c.

1,2 Step 4. An → A strongly in A . From step 1 (i) and (4.1), we have

2 2 kFAn − FAkL = hd(An − A),d(An − A)i ZX

(4.5) = hdAn,d(An − A) + hdA,d(An − A)i ZX ZX An i = − Imh∇i φn, φnihe ,An − Ai + o(1) ZX

To use step 3, we decompose X as Ωn and X\Ωn. On X\Ωn, |φn| ≤ 1. So it is clear that

An i − Imh∇i φn, φnihe ,An − Ai ZX\Ωn 2 (4.6) ≤ (|∇φn| + |An|)|An − A| ZX\Ωn 2 2 1/2 2 ≤( (|∇φn| + |An| ) |An − A| ZX ZX 12 JURGEN¨ JOST, XIAOWEI PENG AND GUOFANG WANG

On the other hand,

An i − Imh∇i φn, φnihe ,An − Ai ZΩn An ≤ |h∇i φn, νni||φn||An − A| (4.7) ZΩn

An 2 1/2 4 4 ≤(− |h∇i , νni| ) kφnkL (kAnkL + kAkL4) ZΩn → 0, as n →∞ (using step 3). (4.5), (4.6) and (4.7) imply

kAn − AkL1,2 → 0, as n →∞.

1,2 + Step 5. φn → φ strongly in L (W ). First, from step 1 (i), we have

2 k∇φn −∇φkL2 = (h∇φn, ∇(φn − φ)i + h∇φ, ∇(φn − φ)i) (4.8) ZX = h∇φn, ∇(φn − φ)i + o(1). ZX

∇φn = ∇An φn − Anφn, so we have

h∇φn, ∇(φn − φ)i ZX

(4.9) = h∇An φn, ∇An (φn − φ)i − hAnφn, ∇(φn − φ)i ZX ZX

− h∇φn,An(φn − φ)i + hAnφn,An(φn − φ)i. ZX ZX From (i) of the Palais-Smale condition, we know

h∇An φn, ∇An (φn − φ)i ZX 1 = − (|φ |2 + s)hφ , φ − φi + dSW (A , φ )(φ − φ) 4 n n n n n n ZX 1 = − (|φ |2 + s )hφ , φ − φi + (s − s)hφ , φ − φi + o(1) 4 n 0 n n 0 n n ZX ZX 1 1 = − (|φ |2 + s )|φ − φ|2 − (|φ |2 + s )hφ, φ − φi + o(1) 4 n 0 n 4 n 0 n ZX ZX (4.10) 2 (since φn → φ strongly in L ) 1 ≤ − (|φ |2 + s )hφ, φ − φi + o(1) 4 n 0 n (since the first term is nonpositive.) 1 ≤ ( ||φ |2 + s |2)1/2 |φ − φ|2 + o(1) 4 n 0 n ZX ZX (since |φ| ≤ s0 by Lemma 3.1) =o(1), VARIATIONAL ASPECTS OF THE SEIBERG-WITTEN FUNCTIONAL 13 as n →∞. The other three terms can be estimated using step 4. For example, the second term

hAnφn, ∇(φn − φ)i ZX

= h(An − A)φn, ∇(φn − φ)i + hAφn, ∇(φn − φ)i ZX ≤kAn − AkL4 kφnkL4 kφn − φkL1,2

+ hA(φn − φ), ∇(φn − φ)i + hAφ, ∇(φn − φ)i ZX ≤kAn − AkL4 kφnkL4 kφn − φkL1,2

+ kφn − φkL2 k∇(φn − φ)kL2 + kAφkL1,2 kφn − φkL2 =o(1).

Therefore, we show that kφn − φkL1,2 → 0 as n →∞. Together with step 4, this completes the proof of the Main Theorem.

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Ruhr-Universitat¨ Bochum, Fakultat¨ fur¨ Mathematik, 44780 Bochum, Germany E-mail address: [email protected]

Ruhr-Universitat¨ Bochum, Fakultat¨ fur¨ Mathematik, 44780 Bochum, Germany E-mail address: [email protected]

Institute of Systems Science, Academia Sinica, 100080 Beijing, China E-mail address: [email protected]